The Implicit Function Theorem: History, Theory, and Applications

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Springer Science & Business Media, Jan 1, 2002 - Mathematics - 163 pages
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The implicit function theorem is part of the bedrock of mathematical analysis and geometry. Finding its genesis in eighteenth century studies of real analytic functions and mechanics, the implicit and inverse function theorems have now blossomed into powerful tools in the theories of partial differential equations, differential geometry, and geometric analysis. There are many different forms of the implicit function theorem, including (i) the classical formulation for C^k functions, (ii) formulations in other function spaces, (iii) formulations for non- smooth functions, (iv) formulations for functions with degenerate Jacobian. Particularly powerful implicit function theorems, such as the Nash--Moser theorem, have been developed for specific applications (e.g., the imbedding of Riemannian manifolds). All of these topics, and many more, are treated in the present volume. The history of the implicit function theorem is a lively and complex story, and is intimately bound up with the development of fundamental ideas in analysis and geometry. This entire development, together with mathematical examples and proofs, is recounted for the first time here. It is an exciting tale, and it continues to evolve. "The Implicit Function Theorem" is an accessible and thorough treatment of implicit and inverse function theorems and their applications. It will be of interest to mathematicians, graduate/advanced undergraduate students, and to those who apply mathematics. The book unifies disparate ideas that have played an important role in modern mathematics. It serves to document and place in context a substantial body of mathematical ideas.
  

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Contents

Preface
ix
Introduction to the Implicit Function Theorem
1
12 An Informal Version of the Implicit Function Theorem
3
13 The Implicit Function Theorem Paradigm
7
History
13
22 Newton
15
23 Lagrange
20
24 Cauchy
27
Variations and Generalizations
93
52 Implicit Function Theorems without Differentiability
99
53 An Inverse Function Theorem for Continuous Mappings
101
54 Some Singular Cases of the Implicit Function Theorem
107
Advanced Implicit Function Theorems
117
62 Hadamards Global Inverse Function Theorem
121
63 The Implicit Function Theorem via the NewtonRaphson Method
129
64 The NashMoser Implicit Function Theorem
134

Basic Ideas
35
32 The Inductive Proof of the Implicit Function Theorem
36
33 The Classical Approach to the Implicit Function Theorem
41
34 The Contraction Mapping Fixed Point Principle
48
35 The Rank Theorem and the Decomposition Theorem
52
36 A Counterexample
58
Applications
61
42 Numerical Homotopy Methods
65
43 Equivalent Definitions of a Smooth Surface
73
44 Smoothness of the Distance Function
78
642 Enunciation of the NashMoser Theorem
135
643 First Step of the Proof of NashMoser
136
644 The Crux of the Matter
138
645 Construction of the Smoothing Operators
141
646 A Useful Corollary
144
Glossary
145
Bibliography
151
Index
161
Copyright

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Page 155 - Graves, Implicit functions and their differentials in general analysis. Transactions of the American Mathematical Society, vol.
Page 157 - WF Osgood, A Jordan curve of positive area, Transactions of the American Mathematical Society, Vol. 4 1903, S.

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